The \(\mathbb {S}^1\)  model allows us to construct geometric maps of real networks by embedding them in the underlying space [1]} using tools such as Mercator [2]}. For visualization purposes, we use the isomorphic and purely geometric formulation of the \(\mathbb {S}^1\)  model, named the m̉odel [3]}. In the m̉odel the hidden degree \(\kappa _i\)  is transformed into a radial coordinate \(r_i\)  in a hyperbolic two-dimensional disk, such that higher degree nodes are placed towards the center and the angular coordinate, indicating similarity, remains the same as in the \(\mathbb {S}^1\)  circle. Remarkably, maps of real networks typically show heterogeneous angular distributions which reveal meaningful communities of similar nodes [4]}, [5]}, [6]}. For further details on both \(\mathbb {S}^1\)  /m̉odels and network embedding, please refer to Methods.
